JUMP PROCESSESGENERALIZING STOCHASTIC INTEGRALS WITH JUMPS

Tyler Hofmeister

University of CalgaryMathematical and Computational Finance Laboratory

Overview

1. General Method

2. Poisson Processes

3. Diffusion and Single Jumps

4. Compound Poisson Process

5. Jump-Diffusion

2016/05/18

Page 1

GENERAL METHOD

General Method

Define aStochasticProcess

Adjust theProcess to aMartingale

Define aStochasticIntegral

Ito’s Formulaand Generator

2016/05/18

Page 3

POISSON PROCESSES

Definition

Definition: Poisson Process

A Poisson process N � {Nt}0≤t≤T ∈ Z+, with intensity λ, is a

stochastic process with the following properties

(i) N0 � 0 almost surely,(ii) Nt − N0 has a Poisson distribution with parameter λt.(iii) N has independent increments, so (s , t)∩ (v , u) � ∅ implies

Nt − Ns is independent of Nv − Nu .(iv) N has stationary increments, so Ns+t − Ns follows the same

distribution as Nt for all s , t > 0.

2016/05/18

Page 5

Poisson Process Example

2016/05/18

Page 6

Poisson Process: Properties

Properties

(i) E [Nt] � λt

(ii) Var [Nt] � λt

(iii) The time between jumps of N are independent and followan exponential distribution.

2016/05/18

Page 7

Compensated Poisson Process

Proposition: Compensated Poisson Process

The compensated Poisson process N �

{Nt

}0≤t≤T

whereNt � Nt − λt is a martingale with respect to it’s generated fil-tration F .

Proof.

E [Nt+s − λ(t + s)|Ft] � E [Nt+s − Ns + Ns − λ(t + s)|Ft]� E [Nt − λt + Ns − λs |Ft]� Nt − λt

�2016/05/18

Page 8

Compensated Poisson Process Example

2016/05/18

Page 9

Stochastic Integral

Definition: Stochastic Integral with respect to aCompensated Poisson Process

Let g be an Ft-adapted process, where Ft is the natural filtra-tion generated by Poisson process N . Define stochastic integralY � {Yt}0≤t≤T of g with respect to N as

Yt �

∫ t

0gs−dNs �

Nt∑k�1

gτ−k −∫ t

0gsλds

where {τ1 , τ2 , . . .} is the collection of times when N jumps.

2016/05/18

Page 10

Ito’s Formula for Poisson Processes

Theorem: Ito’s Formula for Poisson Processes

Suppose Y is the stochastic integral given previously. LetZ � {Zt}0≤t≤T with Zt � f (t ,Yt) for some function f , oncedifferentiable in t. Then

dZt � (∂t f (t ,Yt) − λgt∂y f (t ,Yt))dt

+�

f (t ,Yt− + gt−) − f (t ,Yt−)� dNt

� {∂t f (t ,Yt) + λ([ f (t ,Yt− + gt−) − f (t ,Yt−)]− gt∂y f (t ,Yt))}dt

+ [ f (t ,Yt− + gt−) − f (t ,Yt−)]dNt

2016/05/18

Page 11

Infinitesimal Generator

Recall that the generator Lt of a process Xt acts on twicedifferentiable functions f as

Lt f (x) � limh↓0

E[ f (Xt+h |Xt � x)] − f (x)h

which is a generalization of a derivative of a function which canbe applied to stochastic processes.

The generator of stochastic integral Y from a Poisson processacts as

L Yt f (y) � λ �[ f (y + gt) − f (y)] − gt∂y f (y)�

2016/05/18

Page 12

DIFFUSION AND SINGLE JUMPS

Sum of Stochastic Integrals

Using the framework developed previously for StochasticIntegrals with respect to diffusion and jumps, we sum thesetwo as follows.

Yt �

∫ t

0fs ds +

∫ t

0gs dWs +

∫ t

0hs−dNs ,

where f , g , h are Ft adapted processes, and filtration F is thenatural one generated by both the Brownian motion W andPoisson process N , which are mutually independent.

2016/05/18

Page 14

Ito’s Formula for Single Jumps and Diffusion

Theorem: Ito’s Formula for Single Jumps and Diffusion

Suppose Y is the stochastic integral given previously. LetZ � {Zt}0≤t≤T with Zt � l(t ,Yt) for some function l, oncedifferentiable in t and twice differentiable in y. Then

dZt � (∂t + ft∂y +12 g2

t ∂y y − λht∂y)l(t ,Yt))dt

+ gt∂y l(t ,Yt)dWt + [l(t ,Yt− + ht−) − l(t ,Yt−)] dNt

���∂t + ft∂y +

12 g2

t ∂y y�

l(t ,Yt)+λ([l(t ,Yt− + ht−) − l(t ,Yt−)] − ht∂y l(t ,Yt)) dt

+ gt∂y l(t ,Yt)dWt + [l(t ,Yt− + ht−) − l(t ,Yt−)]dNt

2016/05/18

Page 15

Generator

The generator of Y acts as

L Yt l(y) � ft∂y l(y)+ 1

2 g2t ∂y y l(y)+λ �[l(y + ht) − l(y)] − ht∂y l(y)�

2016/05/18

Page 16

COMPOUND POISSON PROCESS

Definition

Definition: Compound Poisson Processes

Let N be a Poisson process with intensity λ and {ε1 , ε2 , . . .} bea set of independent identically distributed random variableswith distribution function F and E[ε] < +∞. A compoundPoisson process J � { Jt}0≤t≤T is given by

Jt �

Nt∑k�1

εk , t ≥ 0

2016/05/18

Page 18

Compound Poisson Process Example

2016/05/18

Page 19

Compound Poisson Process: Properties

Properties

(i) E [Jt] � λtE[ε](ii) Var [Jt] � λtE

�ε2�

(iii) As with the standard Poisson process, the inter-arrivaltimes are independent and exponentially distributed.

2016/05/18

Page 20

Compensated Compound Poisson Process

Proposition:

The compensated compound Poisson process J �

{Jt

}0≤t≤T

where Jt � Jt − E[ε]λt is a martingale.

Proof.

E[Jt+s |Ft

]� E

[Σ

Nt+sk�1 εk − λ(t + s)E[ε]|Ft

]

� E[Σ

Ntk�1εk + Σ

Nt+sk�Nt+1 − λ(t + s)E[ε]|Ft

]

� ΣNtk�1 − λtE[ε]

�2016/05/18

Page 21

Compensated Compound Poisson Process

2016/05/18

Page 22

Corresponding Stochastic Integral

Let F be the natural filtration generated by J. We define thestochastic integral Y � {Yt}0≤t≤T of an F -adapted process gwith respect to the compensated compound Poisson process Jas

Yt �

∫ t

0gs−d Js �

∑s≤t

gs−∆Js −

∫ t

0gsλE[ε]ds

where ∆Js � Js − Js−

2016/05/18

Page 23

JUMP-DIFFUSION

Sum of Stochastic Integral

Let f , g , and h be F -adapted stochastic processes where F isthe natural filtration generated by an independent Brownianmotion W and J. We define the stochastic integral Y as

Yt �

∫ t

0fs ds +

∫ t

0gs dWs +

∫ t

0hs−d Js

2016/05/18

Page 25

Ito’s Formula for Jump-Diffusion

Theorem: Ito’s Formula for Jump-Diffusion

Suppose Y is the stochastic integral given previously. LetZ � {Zt}0≤t≤T with Zt � l(t ,Yt) for some function l, oncedifferentiable in t and twice differentiable in y. ThendZt � (∂t + ft∂y +

12 g2

t ∂y y − λE[ε]ht∂y)l(t ,Yt))dt

+ gt∂y l(t ,Yt)dWt +�l(t ,Yt− + εNt ht−) − l(t ,Yt−)� dNt

���∂t + ft∂y +

12 g2

t ∂y y�

l(t ,Yt)+λ(E[l(t ,Yt− + ht−) − l(t ,Yt−)] − E[εt]ht∂y l(t ,Yt)) dt

+ gt∂y l(t ,Yt)dWt + [l(t ,Yt− + εNt ht−) − l(t ,Yt−)]dNt

2016/05/18

Page 26

Generator

The generator of Y acts as

L Yt l(y) � ft∂y l(y) + 1

2 g2t ∂y y l(y)

+ λ�E[l(t , y + εht) − l(t , y)] − E[ε]ht∂y l(t ,Yt)�

2016/05/18

Page 27

References

Alvaro Cartea, Sebastian Jaimungal, and Jose PenalvaAlgorithmic and High-Frequency TradingCambridge University Press, 2015

Nicolas PrivaultNotes on Stochastic FinanceNanyang Technological University

2016/05/18

Page 28

Thank you!